Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 241303, 11 pages

doi:10.1155/2012/241303

Research Article

On the Dynamics of the Recursive Sequence

Mehmet Gümüş,¹ Özkan Öcalan,² and Nilüfer B. Felah²

¹Department of Mathematics, Faculty of Science and Arts, Bülent Ecevit University, 67100 Zonguldak, Turkey

²Department of Mathematics, Faculty of Science and Arts, ANS Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey

Received 5 July 2012; Accepted 13 September 2012

Academic Editor: M. De la Sen

Copyright © 2012 Mehmet Gümüş et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We investigate the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation , where , , and the initial conditions are arbitrary positive numbers. We investigate the boundedness character for . Also, we investigate the existence of a prime two periodic solution for is odd. Moreover, when is even, we prove that there are no prime two periodic solutions of the equation above.

1. Introduction

Our aim in this paper is to study the boundedness character, the oscillatory, and the periodic character of positive solutions of the difference equation

where , is a positive, and the initial conditions are arbitrary positive numbers. Equation (1.1) was studied by many authors for different cases of .

In [1] the authors studied the global stability, the boundedness character, and the periodic nature of the positive solutions of the difference equation

where is positive, and the initial values are positive numbers (see also [2–4] for more results on this equation).

In [5] the authors studied the boundedness, the global attractivity, the oscillatory behaviour, and the periodicity of the positive solutions of the difference equation

where , are positive, and the initial values are positive numbers (see also [6–8] for more results on this equation).

In [9] the authors studied general properties, the boundedness, the global stability, and the periodic character of the solutions of the difference equation

where are positive, , and the initial values are positive numbers.

In [10, 11] the authors studied the boundedness, the persistence, the attractivity, the stability, and the periodic character of the positive solutions of the difference equation

where , , are positive, and the initial values , are positive numbers.

Finally in [12, 13] the authors studied the oscillatory, the behaviour of semicycle, and the periodic character of the positive solution of the difference equation

where and , under the initial conditions are positive numbers.

There exist many other papers related with (1.1) and on its extensions (see [14–16]).

Motivated by the above papers, we study of the boundedness character, the oscillatory, and the periodic character of positive solutions of (1.1).

In this paper, also we investigate the case , and of (1.1) and we give a correction for [2, Theorem  2.5].

We say that the equilibrium point of the equation

is the point that satisfies the condition

A solution of (1.1) is called nonoscillatory if there exists such that either

or

A solution of (1.1) is called oscillatory if it is not nonoscillatory. We say that a solution of (1.1) is bounded and persists if there exist positive constant and such that for .

The linearized equation for (1.1) about the positive equilibrium is

2. Semicycle Analysis

A positive semicycle of a solution of (1.1) consists of a “string” of terms all greater than or equal to , with and , such that

A negative semicycle of a solution of (1.1) consists of a “string” of terms all less than , with and , such that

Lemma 2.1. Let be a solution of (1.1). Then either consists of a single semicycle or oscillates about equilibrium with semicycles having at most terms.

Proof. Suppose that has at least two semicycles. Then there exists such that either or . We assume that the former case holds. The latter case is similar and will be omitted. Now suppose that the positive semicycle beginning with the term has terms. Then and so the case

holds for every , , from which the result follows.

3. Boundedness and Global Stability of (1.1)

In this section, we consider the case with no restriction on other parameters and we consider the case with some specified conditions. For these cases, we have the following results which give a complete picture as regards the boundedness character of positive solutions of (1.1).

Theorem 3.1. Suppose that

then every positive solution of (1.1) is bounded.

Proof. We have

Hence we will prove that is bounded. If

then we have , . Hence, the function is increasing and concave. Thus, we get that there is a unique fixed point of the equation . Also the function satisfies the condition

Using [15, 2.6.2] we obtain that is a global attractor of all positive solutions of (1.1) and so is bounded, from which the result follows.

Now we study the boundedness of (1.1) for the case . We give better result than Theorem 3.1 for the boundedness of (1.1) and we prove that in this case, there exist unbounded solutions of (1.1).

Theorem 3.2. Consider (1.1) and assume that , , and . Then every positive solution of (1.1) is bounded and .

Proof. Let

Suppose on the contrary that every positive solution of (1.1) is unbounded. Then, from (1.1), we obtain for . Therefore we get

Thus the proof is complete. We note that in here is a continuous function for .

Theorem 3.3. Consider (1.1) when the case is odd and suppose that

then there exists unbounded solutions of (1.1).

Proof. Let be a solution of (1.1) with initial values such that

Then from (1.1), (3.7), and (3.8) we have

Also, from (3.8) and (3.10)–(3.14), it is clear that

Moreover from (1.1) and (3.8)–(3.14) and arguing as above we get

Therefore working inductively we can prove that for ,

which implies that

So is unbounded. From which the result follows.

Now, in the next theorem, we will provide an alternative proof for the theorem above when and is odd, whose proof can be used for some practical applications.

Theorem 3.4. Consider (1.1) when the case is odd and suppose that . If , then there exists solutions of (1.1) that are unbounded.

Proof. We assume that and choose the initial conditions such that

So,

Therefore, we obtain and . By induction, for , we have and . Thus,

Now, we assume that and choose the initial conditions such that

So, we have

Further, we have

By induction for , we have and . Thus,

from which the result follows.

The following result is essentially proved in [10, 11] for . The result is satisfied for and its proof is omitted.

Lemma 3.5. If Either

or

holds, then (1.1) has a unique equilibrium point .

The following result is essentially proved in [10, 11] for . It is clear that the result is satisfied when is odd and its proof is omitted.

Lemma 3.6. Consider (1.1) when the case is odd. Suppose that

or

holds. Then the unique positive equilibrium of (1.1) is globally asymptotically stable.

4. Periodicity of the Solutions of (1.1)

In this section, we investigate the existence of a prime two periodic solution for is odd. Moreover, when is even, we prove that there are no positive prime two periodic solutions and lastly, we give a correction for Theorem 2.5 which was given in [2].

The following result is given when the case in [10]. If is odd, the result is still satisfied and its proof is omitted.

Lemma 4.1. Assume that k is odd. Then, (1.1) has prime two periodic solutions if and only if

and there exists a sufficient small positive number , such that

Now, let consider the case where is even.

Theorem 4.2. Consider (1.1) when the case is even and the following conditions are satisfied separately:

Then, there are no positive prime two periodic solutions of (1.1).

Proof. Firstly, we consider the case and is even of (1.1) and suppose that

where , is a prime two periodic solution of (1.1). Then it must be

Substituting (4.6) into (4.5), it follows that

Taking logarithm on both sides of (4.7), we obtain that

So from (4.8)

From , thus , which implies that is a unique solution of (1.1).

We consider the case and is even of (1.1). The proof of this case is similar to the first case’s proof and will be omitted.

Now, suppose that , and is even of (1.1). In this case we have that

Considering the numerator on the right hand side in (4.10) let

From and ,

which implies that is a unique solution of (1.1).

Suppose that , and is even of (1.1). The proof of this case is similar to the third case’s proof and will be omitted.

The following result was given in [2, Theorem 2.5] for (1.1) when the case , and . But the authors make some mistakes in this theorem. Now, we give a correction and a conjecture for this result.

Theorem A. Consider (1.1). Let be , , , and . Suppose that

hold. Then the unique positive equilibrium of (1.1) is globally asymptotically stable.

Correction B

Consider (1.1). Let be , , , and . Suppose that

holds. Then the unique positive equilibrium of (1.1) is globally asymptotically stable.

Proof. It is easy to see the proof from Theorem  2.5 in [2].

Conjecture 4.3. Consider (1.1). Let be , , , and . Suppose that

holds. Then the unique positive equilibrium of (1.1) is globally asymptotically stable.

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